User blog:Chronolegends/Egg Notation
Egg Notation, Red and Blue eggs Motivation: The idea was to make a FGH clone without using math language like functions, iterations, ordinals, etc.. in order to make it the least mathy possible it i chose to make it completely without numbers Why eggs?, because notations in googology sooner or later end up being mixed into salad numbers so i figured i would provide a tasty ingredient for any salad. :; Concepts ; Eggs A pair of grouping symbols ; Egg colors : Red = () : Blue = [] ; Egg groups : Contiguous Eggs ; Syntax Rules :There may be no red egg to the left of a non-red egg. Example: ()[] is invalid :The above rule also applies to eggs inside other eggs :There is at least one red egg ; Symbol Glossary : R = Zero or more Red Eggs. : B = Zero or more Blue Eggs. : A/B = Zero or more eggs of colors A and/or B : O = Other eggs. : Xa, Xb = Different instances of X, for example Ri = ()() , Rj = ()()()() :cX = Counts the X eggs. Example: R = ()()()() , cR = 4 :eX = Apply the reduction rules on X :x...y = Create y copies of x. Egg copies have the same contents as the original ; Rules I added a text-only explanation for the folks who aren't used to reading math-like formulae : Loop Rule: Crack the rightmost non-red egg, then apply the appropiate rule, depending on the color and contents : Reduction rules 1: O[]R → O()R :Description: Cracked empty Blue Eggs are replaced with one Red Egg 2: O()RbRa → ORb...cRa Ra :Description: Cracked Blue Eggs with one or more Red Egg inside are replaced with an amount of Blue Eggs equal to the Red Eggs to the right of the cracked egg, the new Blue Eggs have the same contents as the cracked egg, except for 1 less Red Egg. 3: OB/RaRb → OeB/RaR :Description: For Blue Eggs inside other Blue Eggs, simply solve the inner eggs using rules 1 and 2 until there are only Red Eggs. Note: Since eB/Ra → Rb , groups with blue / red eggs are interchangeable with groups having only red eggs, this means cases with B/R are accounted for by just writing R : Termination rule: If there are only red eggs, stop ; A function to extract a number from Egg Notation : ceX, where X = a bunch of eggs ;Egg Notation, Green Eggs, Part 1 :New concept: Egg whites, separation between egg groups inside eggs, represented by a comma , :New egg color: Green = {} :New symbol: G = Zero or more Green Eggs :New symbol: →...n repeat this replacement n times. :New symbol: {...,X...} X is nested inside zero or more Green Eggs, right group :New symbol: {...X,...} X is nested inside zero or more Green Eggs, left group :New sytnax rule: A Blue Egg may contain Eed Eggs or Green Eggs, but not both. :New syntax rule: Green Eggs must be inside a Blue Egg or a Green Egg :New syntax rule: Only one eggwhite is allowed per Green Egg. :New syntax rule: Only Red Eggs may be contained in the group to the left of an Eggwhite :New reduction rules: 4: O{...,{,}...}R → O{...,R...}R :Description: When the inner green egg is empty, replace it with as many Red Eggs as touch the Blue egg 5: O{...,{Ra,()Rb}...}Rc → O{...,{Ra,Rb}...}...cRc Rc :Description: When the inner green egg has red eggs in the right group, remove one of those red eggs, then copy the resulting blue egg, a number of times equal to the red eggs touching it. 6: O{...,{()Ra,}...}Rb →...cRb O{...,{Ra,X}...}Rb where X = {Ra,X}, except the last replacement is {Ra,} :Description: When the inner green egg has no red eggs in the right group, remove one red egg from the left group, then copy the resulting egg in the right group, and in the right group of the egg copy, repeating for a total number of copies equal to the amount of Red Eggs touching the Blue Egg. ;Egg Notation, Green Eggs, Part 2 Eliminates this rule. "New syntax rule: Only red eggs may be contained in the group to the left of an eggwhite" Modifies these reduction rules : 4: O{...,{...{,},...}...}}R → O{...,{...R,...}...}}R :Description: Same as before, but applies to the inner right egg of the inner left egg 5: Oa{...,{Ob,()Ra}...}Rb → Oa{...,{Ob,Ra}...}...cRb Rb :Description: Same as before but the left group may contain green eggs 6: O{...,{...{()Ra,},...}...}Rb →...cRb O{...,{...{()Ra,X},...}...}Rb where X = {Ra,X}, except the last replacement is {Ra,} :Description: Same as before but, applies to the inner right egg of the inner left egg : New reduction rule!: 7: Oa{...,{...{Ob,()Ra},...}...}Rb →...cRb O{...,{...{Ob,X},...}...}Rb where X = {Ra,X}, except the last replacement is {Ob,Ra} :Description: When the inner Green Egg has some Red Eggs in the right group, remove one Red Egg from the right group, then copy the resulting egg in the right group, and in the right group of the egg copy, repeating for a total number of copies equal to the amount of Red Eggs touching the Blue Egg. Category:Blog posts